Questions tagged [cauchy-schwarz-inequality]

The Cauchy-Schwarz inequality states $|\langle x,y \rangle |\leq ||x||\cdot ||y||.$ Use this tag for questions related to the CS inequality and its applications.

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Inequality involving random vectors and absolute values

Let $\mathbb{X}, \mathbb{Y} \subset \mathbb{R}^d$ be finite sets. Suppose random vectors $X \in \mathbb{X}$ and $Y \in \mathbb{Y}$ are sampled according to a joint distribution $\mathbb{P}_{XY}$. ...
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Reverse Inequality

I was doing some numerical integration when I figured the function I was dealing with (i.e., the function I was integrating) evaluated to big numbers on a tiny portion of the interval (over which I ...
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Good upper-bound for $\mathbb E[|X-np|^r]$ where $X \sim \text{Binomial}(n,p)$ and $r \ge 1$

Disclaimer. Question moved from SE. Setup Let $X \sim \text{Binomial}(p, n)$, and $r \ge 1$. Question What is a good upper-bound for $\mathbb E[|X-np|^r]$ ? Solution for small $r$ If $r=2$, then ...
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Bounds for $\sum_{t=1}^Tn_t(s_t)^{-\alpha}\mu(s_t)$ where $n_t(s) = \sum_{1 \le t' \le t} 1_{\{s_{t'}=s\}}$ for $s \in [k]$ and $\mu \in \Delta_k$

Disclaimer: I'm not certain this is the right venue for this post, but I'll give it a try... So trying prove some bounds in my ongoing work in theoretical reinforcement learning, I encountered the ...
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Is there a tight lower bound for the expectation of the product of two positive valued random variables?

Let $X,Y$ be two (dependent) random variables with $\mathbb{P}(X\ge 0)=\mathbb{P}(Y\ge 0)=1$. I want to find a tight lower bound of $\mathbb{E}(XY)$ when $X,Y$ are non-negative, almost surely. ...
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Is it possible to find $f$ such that : $f$ is absolutely integrable, $f'$ is absolutely integrable and such that $f$ is not $1/2$-Hölder

I am trying to find a function $f: \mathbb{R}^+ \to \mathbb{R}^+$ that fullfils the following conditions $$f \in \mathcal{C}^1(\mathbb{R}^+,\mathbb{R}^+)$$ $$\int_{\mathbb{R}^+} f \in \mathbb{R}^+$$ ...
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On the Cauchy-Schwarz Inequality for trace function of random matrices

In the deterministic case, for two matrices $A$ and $B$ with appropriate matrices, we know that $$tr((A^{T}B)^{2})\leq tr(A^{T}A)tr(B^{T}B)$$ which is the trace form of Cauchy-Schwarz-Inequality (CSI)....
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A probabilistic angle inequality

Conjecture: There is a universal constant $c$ such that for any fixed nonzero real vector $q$ of any dimension $n$ and any random vector $p$ of the same dimension $n$ with independent components ...
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An inequality involving a sum of power terms

I am currently working in a problem in Information Theory and I came across a difficult inequality. After many attemps, I simplified the inequality, which now looks at follows. Consider a positive ...
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An alternative proof of Bayesian Cramer-Rao

My question is: Are there an alternative proof of Cramer-Rao lower bound that does not use Cauchy-Swartz inequality? Let me outline the classical proof and explain why I am interested in this ...
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Does this simple inequality have a name?

Let $x_{1},\ldots,x_{n}$ be nonnegative numbers such that $m \leq x_{i} \leq M$. Let $$S=\sum_{i=1}^{n}{x_{i}}$$ and $$Q=\sum_{i=1}^{n}{x_{i}^{2}}.$$ Then $$Q \leq S(M+m)-nMm.$$ This has ...
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An Linear Algebra Inequality

How to prove the following inequality: Let $X$ and $Y$ be $n\times m$ matrices with real entries. Prove that \det\left(XY^T\right)^2 \leq \det\left(XX^T\right)\det\left(YY^T\right) . \...
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A plausible inequality

I come across the following problem in my study. Let $x_i, y_i\in \mathbb{R}, i=1,2,\cdots,n$ with $\sum\limits_{i=1}^nx_i^2=\sum\limits_{i=1}^ny_i^2=1$, and $a_1\ge a_2\ge \cdots \ge a_n>0$. Is ...
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Is there a combinatorial proof of Cauchy-Schwarz?

I've only played with this a little for the past day or so, and haven't thought about it too hard, so it might be obvious. Obviously it's not fair to ask for a "combinatorial proof" of an inequality ...
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